# Shear jamming and fragility in dense suspensions

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## Abstract

The phenomenon of shear-induced jamming is a factor in the complex rheological behavior of dense suspensions. Such shear-jammed states are fragile, i.e., they are not stable against applied stresses that are incompatible with the stress imposed to create them. This peculiar flow-history dependence of the stress response is due to flow-induced microstructures. To examine jammed states realized under constant shear stress, we perform dynamic simulations of non-Brownian particles with frictional contact forces and hydrodynamic lubrication forces. We find clear signatures that distinguish these fragile states from the more conventional isotropic jammed states.

## Keywords

Shear jamming Suspension rheology Granular physics## 1 Introduction

Suspensions, in which solid particles are dispersed in a viscous liquid, are a class of complex fluids found frequently in nature, industry, and consumer applications [1, 2]. To predict flows of suspensions with arbitrary macroscopic boundary conditions, it is necessary to develop continuum models based on particle-scale physics; it is too expensive to simulate individual motions of particles and interstitial flows for macroscopic problems. Dilute suspensions, in which the solid volume fractions \(\phi\) are less than about 5%, are well described with the Newtonian constitutive model with a modified viscosity [3, 4]. However, constitutive models for denser suspensions exhibiting more complex rheological properties are still not available [5, 6].

*Fragile matter*as a constitutive class of complex fluids was introduced to describe emergence of rigidity in flowing suspensions [10, 11]. First, it is helpful to emphasize that there are only three possible states in non-Brownian suspensions of rigid particles regarding mechanical responses (see Fig. 1a):

- (i)
*liquid-like*state, that cannot statically bear any shear stress. - (ii)
*solid-like*state, that can statically bear stresses in all directions. - (iii)
*fragile*state, that can statically bear stresses only within a certain range of directions.

A shear-jammed state is one which is reached by shear but is then able to statically bear the shear stress (or ‘load’). As this state is not statically stable for applied stresses incompatible with the jammed state, it is termed fragile in the terminology of Cates et al. [11], and we follow that terminology here; Fig. 1a (iii) illustrates the concept. Upon reversal of the shear stress from a fragile shear-jammed state, the suspension will flow, i.e., undergo some finite strain, before possibly reaching a jammed state in the new direction.

The occurrence of shear jamming under quasi-statically imposed strain was experimentally elucidated in frictional grains by Bi et al. [13], and its connection to Reynolds dilatancy was elaborated in Ren et al. [14]. Two types of shear-jammed states were identified at a given \(\phi\): states created at lower strains, which could not sustain shear reversal, and states at strains higher than a characteristic value, which could. In [13], the former were referred to as fragile and the latter as shear-jammed. Sarkar et al. [15, 16, 17] developed a theoretical framework to describe the transition between the two types of jammed states identified in [13]. Recent numerical work by Otsuki and Hayakawa [18] showed that this transition could be detected through imposition of oscillatory shear. It should be noted that those previous studies on shear jamming were performed with strain-controlled protocols, in which deformation is forced regardless of whether jammed or not. In contrast, we investigate shear jamming with a stress-controlled protocol; once a system is jammed, no further deformation occurs in the same direction. This stress-controlled approach seems, in this way, more natural to capture shear jamming than the previous works. (Very recently, Srivastava et al. [19] also investigated “shear-arrested states” using constant-stress discrete element simulations.) In this article, we examine fragility of shear jammed states by performing particle dynamics simulations with idealized conditions: inertialess, non-Brownian, and monolayer systems.

## 2 Simulation model

We consider suspended particles in a viscous liquid. The particles are sufficiently small for all inertial effects to be negligible, i.e., the Stokes number \(St \equiv \rho a^2 {\dot{\gamma }}/\eta _0\) (with \(\rho\) and *a* being the density and radius of particles, respectively, \({\dot{\gamma }}\) being the shear rate, and \(\eta _0\) being the viscosity of the suspending fluid) is vanishingly small. We also omit Brownian motions, which are relevant for smaller particles. Stokesian Dynamics (SD) is an efficient method to reproduce particle dynamics in this Stokes regime [20]. The viscosity divergence predicted by the original SD is a dynamic effect due to the singularity of hydrodynamic lubrication [21, 22]. Recently, the SD approach (with only hydrodynamic lubrication) was extended to be coupled with frictional contact mechanics to reproduce discontinuous shear thickening [23]. In this work, we employ an algorithm to mimic stress-controlled rheology [24]. The viscosity divergence under a constant shear stress is just a consequence of a static force balance of the contact forces. In contrast to the original SD, the hydrodynamic contributions vanish at the viscosity divergence. Therefore, the results shown in this article would share some common features with dry granular systems in the quasi-static limit.

### 2.1 Stress-controlled quasi-static dynamics

*xy*component of (3), we can determine \({\dot{\gamma }}\) for the given shear stress \(\sigma ^{xy}\),

### 2.2 Contact force model

In many stable suspensions used in practice, some repulsive forces act between non-contacting particles, preventing flocculation due to short-range van der Waals attractions. However, in this article, we focus on a simple model system in which the non-hydrodynamic interaction consists only of contact forces: \(\varvec{F}_{\mathrm {nh}} = \varvec{F}_{\mathrm {c}}\).

To model \(\varvec{F}_{\mathrm {c}}\), we employ a soft-constraint approach. The first geometrical constraint is the volume excluding force of solid particles. Hard-sphere particles will have zero overlap. To mimic this, we introduce a harmonic penalty function \((k_{\mathrm {n}}/2)(a_i+a_j-r_{ij})^2\), which generates a force along the normal direction. Here, \(a_i\) and \(a_j\) are radii of particles *i* and *j*, and \(r_{ij}\) the distance between them.

When rough or bumpy solid surfaces are in contact, their sliding displacements are also restricted by friction or interlocking; here, we introduce another harmonic penalty function \((k_{\mathrm {t}}/2)\varvec{\xi }^2\) of the relative sliding displacement \(\varvec{\xi }\), which is determined with translations and rotations of contacting particles [30]. This generates tangential forces acting at the contact point. Regarding the maximum tangential force, we employ a simple Coulomb friction law, where the upper bound is proportional to the normal force with a proportionality coefficient \(\mu\). In this work, we mainly study an infinite friction coefficient, implying that sliding displacements are constrained as long as the particles are pushed inward.

This soft-constraint approach is fundamentally different from hard-sphere algorithms, which impose strict geometrical constraints. To make the constraints stricter, we would need to set sufficiently large values for the penalty parameters \(k_{\mathrm {n}}\) and \(k_{\mathrm {t}}\). The values which we selected keep the maximum overlap and tangential displacement less than 2% of the particle radius below the isotropic jamming point.

## 3 Results and discussion

We study monolayer bidisperse systems of 1000 spherical particles, with a size ratio of 1.4 and a volume ratio of approximately 1. To generate initial configurations, we used Brownian simulations to relax randomly placed particle configurations.

### 3.1 Shear reversal test

We start by confirming the concept of fragile matter with our simulation model for dense suspensions. To understand the roles of shear-induced structure, Gadala-Maria and Acrivos [31] performed shear reversal tests using a rate-controlled setup. Here, we simulate a stress-controlled shear reversal test.

The system is shear-jammed when all particles are in static force balance \(\varvec{F}_{\mathrm {C}}^{(i)} = 0\) and a contact network to support all stress is formed such that \(\sigma ^{xy}_{\mathrm {C}} = \sigma ^{xy}\). According to (1) and (6), these conditions lead to \(\varvec{U}^{(i)} = 0\) and \({\dot{\gamma }} = 0\). We consider states to be jammed with the following criteria: \(\max |\varvec{V }^{(i)}| < 10^{-3}a {\dot{\gamma }}_0\) and \(|{\dot{\gamma }}| < 10^{-8} {\dot{\gamma }}_0\); the characteristic shear rate \({\dot{\gamma }}_0 \equiv \sigma '/ \eta _0\) is used. Here, \(\varvec{V}^{(i)} \equiv \varvec{U}^{(i)}-\varvec{u}(\varvec{r}^{(i)})\) are non-affine particle velocities. \(\gamma _{\mathrm {J}}^{(1)}\) denotes the total strain to the shear-jammed (SJ) states from the relaxed initial configuration. Jamming occurs above the isostatic condition \(Z > Z_{\mathrm {iso}}^{\mu =\infty } = 3\) [32], where *Z* is the average contact number with non-rattlers. Particles that have fewer than two contacts with non-rattlers are called rattlers, and thus we need some iteration to determine them. As seen in Fig. 2d, the isostatic condition does not immediately lead to the SJ state.

Now, we reverse the shear stress to \(\sigma ^{xy}= -\sigma '\), corresponding to a rotation of the principal stress axes by \(\pi /2\). Since the previously formed contact network cannot support this new stress, the suspension unjams. The contact number drops to a value \(Z_{\mathrm {min}}\), being below the isostatic condition (Fig. 2d). Thus, the stress-reversal simulation confirms that the SJ state is *fragile*. We continue the simulation with \(-\sigma '\). The particle dynamics is not reversible, and the state does not return to the initial configuration; rather, it reaches another SJ state after strain \(\gamma _{\mathrm {J}}^{(2)}\).

After reaching the second jammed state, we stop applying the stress \(\sigma ^{xy} = 0\) to confirm the smallness of the elastic recovery strain (Fig. 2b). This small recovery is due to the finite values of the penalty parameters \(k_{\mathrm {n}}\) and \(k_{\mathrm {t}}\) in the soft-constraint contact model. In the ideal hard-sphere limit, the recovery strain will be zero. If stress is applied in the same direction again, the system will not flow because the contact network remains.

### 3.2 Features of shear-jammed states

The SJ state is realized due to formation of a stress-bearing contact network. The particle movements obey the force and torque balance equations (1), and rearrangements continue until static balances are globally achieved. The structural evolution to reach the global balance is not monotonic. As seen in the movie in Supplementary Material, a static force balance, which is roughly indicated by vanishing velocities (dark colors), is locally achieved in advance of other parts, but the local stress axes may change due to rearrangements of other parts. Thus, the local domains, which achieved a force balance once, need to be restructured again (cf. “micro-fragility” in [10]).

Though the isostatic condition \(Z = Z_{\mathrm {iso}}\) (\(= 3\) for \(\mu = \infty\)) alone does not determine whether or not shear jamming occurs, shear jamming was realized at slightly larger \(Z \approx 3.1\) in all of our simulations for \(\phi \le 0.84\) (Fig. 3b). If we were to run more simulations with larger \(\gamma _{\mathrm {max}}\), the minimum line of *Z* might approach the isostatic condition. (The SJ state at the lowest possible \(\phi\) may be close to random loose packing [33, 34], but is anisotropic owing to the shearing by which it is accessed.) We can confirm fragility with the minimum value (red downtriangle) of *Z* after the stress reversal. If *Z* goes below \(Z_{\mathrm {iso}}\), the system must experience liquid-like states no matter how short their duration.

### 3.3 Isotropic jamming

Strains to achieve jammed states become progressively lower at higher area fractions (Fig. 3a). The strains \(\gamma _{\mathrm {J}}^{(1)}\) are less than 0.01 at \(\phi \ge 0.85\), and higher penalty parameters for the contact model, \(k_{\mathrm {n}}\) and \(k_{\mathrm {t}}\), can make them even lower (data are not shown). The vanishing value of \(\gamma _{\mathrm {J}}^{(2)}\) suggests that the state does not flow in any direction, i.e., the jammed state is solid-like. As seen in Fig. 3c, the stresses of these states indeed become more isotropic (The ratio \(\sigma ^{xy} / P\) is one way to represent the stress anisotropy, where *P* is the particle pressure [35]). We can also see the sudden increase in *Z* above \(\phi = 0.85\) (Fig. 3b). These observations suggest that the solid-like jammed states are obviously distinguishable from the SJ state, but similar to the conventional isotropically jammed state despite the presence of friction. The transition point \(\phi _{\mathrm {IJ}}\) seems to be in the range \(0.84< \phi _{\mathrm {IJ}}< 0.85\), which agrees with the known value (about 0.84) for frictionless 2D systems [36].

### 3.4 Stress-bearing structure

### 3.5 Friction

*Z*are required to realize jamming (Fig. 5b). We also notice that the contact number

*Z*of unjammed states increases with \(\mu\); particles tend to contact in frustrated flows due to friction.

*Z*of jammed states drop to below the isostatic condition (\(Z_{\mathrm {iso}}^{\mu =0} = 2d\)) after the stress reversal (Fig. 6b). However, this occurs in a narrow range just below the isotropic jamming. Our particles seem too soft to see a clear transition from the shear jamming to the isotropic jamming. As discussed elsewhere [37], the shear jamming observed in frictionless particles can be just due to a finite size effect. Thus, our current simulation cannot confirm the existence of shear jamming without friction.

## 4 Conclusions

We confirmed that dense suspensions with frictional interactions between particles can behave as fragile matter. In a flowing dense suspension under stress, a contact network is formed. The suspension becomes jammed when the shear-induced contact network statically supports the entire stress. However, this jammed state is not stable; a change of stress axes makes it flow. This fragile instability is the most important feature of shear jamming to be distinguished from the solid-like isotropic jamming. Furthermore, we found various signatures to distinguish the two different states in the average contact number *Z*, the drop of *Z* after the stress reversal, and the stress anisotropy \(\sigma ^{xy}/P\). It is also worth noting that SJ states near the lower bound are truly “fragile.” We need to set a sufficiently short time step to capture such SJ states in simulations with the soft-constraint contact model.

In this article, we did not investigate the dependence on the strength \(|\sigma ^{xy}|\) of the shear stress. Ideal inertialess hard-sphere suspensions do not possess any internal force scale; thus, the states must be independent of the stress scale. Therefore, in the phase diagram with stress and area fraction, shear jamming lies in the vertical boundaries: \(\phi _{\mathrm {SJ}}< \phi < \phi _{\mathrm {IJ}}\). If some interparticle repulsive forces or Brownian forces act on particles, they tend to hinder the formation of a contact network. This competition introduces a stress dependence. Deformability of particles also causes a similar stress dependence; contact deformation can enhance tangential constraints restricting sliding and rolling degrees of freedom [11]. It is worth noting the distinction between shear thickening and Reynolds dilatancy here [38]. Shear thickening does require such an internal force scale besides tangential constraints; this makes the rheology of suspensions rate dependent [23]. Shear jamming is relevant to shear thickening but is a more basic phenomenon; it can occur just due to shear strain without the internal force scale, as demonstrated in this article. Since the volume of a suspension is constrained, shear jamming of dense suspensions can be considered as a confined Reynolds dilatancy [39].

## Notes

### Acknowledgements

The authors would like to thank M. Otsuki, H. Hayakawa, and R. Mari for fruitful discussions. This study was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grants Nos. 17H01083 and 17K05618. BC was supported by NSF-CBET-1605428, while JFM was supported by NSF-CBET-1605283. The research was also supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. RS thanks R. Yamamoto for his full support.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## Supplementary material

Supplementary material 1 (m4v 12677 KB)

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